Ok, to be clear, let’s say I’m dumb and accidentally go with the default (I get the color of the opt out button wrong or something). As if there’s a “publish my private emails to the internet” default-on button in email. Then, I use it to edit a rec letter for student X, with my signature Y. (Yes I know this is dumb and I try changing names when editing but am sure some actual names may slip through.) A few months later the next model is released trained on the data. Student X asks Claude what Y would write in a rec letter about X. Such a button is a “wings stay on / wings fall off” button on a plane.
You're severely overestimating the ability of the model to recall a single mostly uninteresting item from it's billions of input documents.